À¶Ý®ÊÓƵ

Faisal Romshoo, University of À¶Ý®ÊÓƵ

Constructing calibrated submanifolds through evolution equations

I will talk about how we can construct examples of calibrated submanifolds using the techniques of evolution equations. We will begin by defining the ideas involved in coming up with these evolution equations and then look at some of the examples of calibrated submanifolds that are constructed this way, following arXiv:math/0008021, arXiv:math/0008155, arXiv:math/0010036 and arXiv:math/0401123.

MC 5403

Xuemiao Chen, University of À¶Ý®ÊÓƵ

On the space of lines

I will make a story about the space of oriented lines in the three dimensional Euclidean space.

MC 5403

Jiahao Hu, Yau Mathematical Sciences Center, Tsinghua University

Homotopy theoretical holomorphic invariants of complex manifolds

In this talk, I will begin by presenting a method for extracting new holomorphic invariants of a complex manifold from its de Rham algebra of complex-valued differential forms. These invariants can be seen as refined versions of the complexified homotopy groups. I will then explore their potential connections with Hermitian geometry. Specifically: (1) the non-abelian part (refined fundamental group) should be related to a generalization of Higgs bundle; (2) the abelian part (refined higher homotopy) may provide new tools for studying the geometry of holomorphic mappings.

MC 5417

Sean Lee, University of À¶Ý®ÊÓƵ

Topological dynamics of the Rado graph

We introduce some concepts from topological dynamics, in particular the universal minimal flow, with the goal of showing that the universal minimal flow of the automorphism group of the Rado graph is the space of linear orders of the Rado graph.

MC 5417

Yash Singh, University of À¶Ý®ÊÓƵ

Vector Bundles on Toric Stacks

We give moduli interpretations of toric vector bundles and generalize this approach to a classification of bundles on arbitrary toric stacks.

MC 5403

Facundo Camano, University of À¶Ý®ÊÓƵ

A Gromov—Hausdorff Convergence Result for the Moduli Space of Singular Monopoles

I will introduce singular monopoles on R^3 and their moduli space. We will then focus on U(2) singular monopoles, which have known explicit expressions, and look at the Gromov—Hausdorff convergence of the moduli space as one singularity is sent off towards infinity.

MC 5403

Gian Cordana Sanjaya, University of À¶Ý®ÊÓƵ

Density of Special Classes of Polynomials with Squarefree Discriminant

In this talk, we discuss the problem of determining the local densities of monic integer polynomials of fixed degree with squarefree discriminant, with some conditions on the coefficients. Previously, Yamamura solved the case where no extra conditions are imposed, and Bhargava, Shankar, and Wang proved that the global density of general monic integer polynomials (of fixed degree) with squarefree discriminant is equal to the product of the local densities.

Our main case of interest is when the first two non-leading coefficients are fixed. However, we also obtain results on other cases, such as the case where the constant coefficient is fixed. Moreover, the technique used here allows us to compute the local densities exactly.

This talk is based on the recent preprint: https://arxiv.org/abs/2505.06820

MC 5417

Andy Zucker, University of À¶Ý®ÊÓƵ

Dynamical Partite Construction

We revisit the partite construction using some of the dynamical ideas we have developed.

MC 5417

Spiro Karigiannis, University of À¶Ý®ÊÓƵ

Unique continuation in geometry (conclusion)

I will finish discussing the paper by Jerry Kazdan on unique continuation in geometry. I will try to make this second talk self-contained, by stating the various estimates which we derived in my first talk, and continuing the proof from there.

MC 5403

Justin Fus, University of À¶Ý®ÊÓƵ

The Geometry of the Based Loop Group and Moment Maps

Given a compact Lie group, we will explore a symplectic structure on the infinite-dimensional based loop group consisting of smooth maps from the circle to the Lie group with the identity as a basepoint. The maximal torus of the Lie group and the circle group together generate a Hamiltonian torus action on the loop group. Results on connectedness of level sets and convexity of the moment map, which are attempts to generalize those for finite-dimensional compact symplectic manifolds, will be previewed.

MC 5403